Pricing Options Using Lattice Rules

There are many examples of option contracts in which the payoff depends on several stochastic variables. These options often can be priced by the valuation of multidimensional integrals. Quasi–Monte Carlo methods are an effective numerical tool for this task. We show that, when the dimensions of the problem are small (say, less than 10), a special type of quasi–Monte Carlo known as the lattice rule method is very efficient. We provide an overview of lattice rules, and we show how to implement this method and demonstrate its efficiency relative to standard Monte Carlo and classical quasi–Monte Carlo. To maximize the efficiency gains, we show how to exploit the regularity of the integrand through a periodization technique. We demonstrate the superior efficiency of the method both in the estimation of prices as well as in the estimation of partial derivatives of these prices (the so-called Greeks). In particular this approach provides good estimates of the second derivative (the gamma) of the price in contrast to traditional Monte Carlo methods, which normally yield poor estimates. Although this method is not new, it appears that the advantages of lattice rules in the context of insurance and finance applications have not been fully appreciated in the literature.